Robert Recorde used dialogue between two characters—master and scholar—very effectively as a teaching/learning device. Galileo, in his 1632 Dialogue Concerning the Two Chief World Systems expanded the dialogue to three characters. Salviati, a brilliant teacher, presented the case for a new theory in which the planets revolve around the sun; Sagredo, an intelligent amateur, served as the intelligent layman who posed reasonable questions; and Simplicio, a good-humoured simpleton, argued the conventional case. Galileo’s mistake was to present the heliocentric case as fact rather than hypothesis in defiance of the directives of the Catholic Church. His second mistake was to too closely match the character of Simplicio to his old friend Maffeo Barberini--Pope Urban VIII.
In 1992, I used this device to create an article Dialogue on the Great Examination System (http://www.bcamt.ca/wp-content/uploads/vector/333-Summer-1992.pdf pp. 48-53). Here Shamus, a university professor, presents the case for a variety of alternatives to standardized testing; Sonia, a secondary mathematics teacher, is the intelligent practitioner; and Slim, a lawyer, is a bit of a good-natured dolt.
In Mathematics Education Across Time and Place
Vitorio de la Francesca, a reckoning master, navigator, and instrument maker in 17th-century Venice (Chapter 3: The Italian Renaissance) describes his experience at university and his initial exposure to the teachings of Galileo.
In Verona, apart from theology, I studied the seven liberal arts: grammar, logic, rhetoric, arithmetic, music, geometry and astronomy. There, for the first time I listened to the lecture about Galileo’s mechanics and his laws of free-falling bodies. A year ago, in 1633, in Rome, Galileo had recanted all his heretic teachings and was again welcomed to the Church of Rome. However, his teachings, though sternly criticised, were openly discussed at the University. My professor of astronomy, a thin, tall Jesuit, refused to pronounce his name and called him “a madman from Padua.” Once, speaking about the teachings of the madman from Padua, he pretended to be swaying, then holding onto a desk he shouted with a frightful sneer, "Hold me, the earth is turning." He mocked Galileo’s theory that the earth revolved around the sun and around itself. He angrily said that everything had started with that Copernicus, a heretic, when we began to calculate from his tables the length of the solar year and the dates of the eclipses of sun and moon. "He dared to deny Ptolemy's system," he shouted. Then he read from the Bible the chapter about the creation of the universe. When he finished, he closed the book with a dramatic gesture and asked in a menacing voice, “Does the Holy Scripture lie?” We waved our heads obediently, trying to show with our facial expression how much we found the theory repugnant. However, knowledge is a strange thing. It is like a genie in a bottle: once out, you cannot put him back. Galileo’s theories were discussed openly, and though everybody expressed stern disagreement, something strange began to happen. People grew suspicious about the truth the church has been preaching for centuries. It seemed that the foundations of the whole structure began to shake. In particular, at the university of Verona, a group of students discussing a reformed astronomy gathered around Torricelli, a follower of Galileo, who had recently obtained a position at the university. At informal gatherings, he told us that nature was mathematical and that the constitution of the universe conformed to simple mathematical relationships. With several of the most trusted disciples he shared the controversial manuscript written by Galileo that he kept in order to protect his teacher. In the Dialogue Concerning the Two Principal Systems of the World, the Ptolemaic and the Copernican Galileo showed that the centrifugal force depended not on the linear velocity of a point on the earth's surface but on the angular velocity of rotation which was the main objection to Copernican theory. I noticed that when he spoke in public he was very cautious to make everything sound as a mere hypothesis. It was through this Tuscan, Torricelli, that I became familiar with the principal works in ancient mathematics and mathematics of more recent times. Still, knowledge and intelligence were dangerous things even at this place.
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